[EM] Continuous geographical proportionality

[Kristofer Munsterhjelm]

Juho wrote:
> On Aug 28, 2008, at 13:18 , Kristofer Munsterhjelm wrote:
> 
>>> One more approach to this would be to provide "perfect" continuous 
>>> geographical proportionality. One would guarantee political and 
>>> geographical proportionality at the same time. One would try to 
>>> minimize the distance to the closest representative from each voter 
>>> and make the number of represented voters equal to all 
>>> representatives. In short, distribution of representatives would be 
>>> close to the distribution of the voters (while still maintaining also 
>>> political proportionality).
>>
>> There would, of course, be limits to the guarantee of having both 
>> political and geographical proportionality at the same time. If your 
>> immediate vicinity have candidates whose opinion you completely 
>> disagree with, one of geographical proportionality and political 
>> proportionality will have to sacrifice part of itself for the other.
> 
> Yes, smaller political groupings would not get as "near" 
> representatives. That is also natural since there are so few of them. It 
> is also possible that close to the voter there are many party A 
> supporters and therefore they get a seat. In the next neighbourhood 
> there are lots of B part supporters, and so on. But probably we would 
> still get a more accurate geographical proportionality than with large 
> districts.
> 
> One seat districts would be geographically very proportional, but your 
> nearest representative of your own party could be far away. In this new 
> model one could try to improve also this (=> geographical 
> proportionality within parties too; or count weights for the distances 
> based on the preferences of individual voters).

That's right. Patching up one-seat districts by using either FMV or MMP 
detrimentally affects the geographical proportionality aspect, since 
either some seats no longer go to the "true" winners (FMV) or 
geographically unrelated top-up seats (MMP), as you said.

>> In the long run, the effect might self-stabilize, if for no other 
>> reason that if there are many Y-ists in an area, one of them is going 
>> to notice and want to become a candidate.
>>
>> I'm not quite sure how to do perfectly continuous geographical 
>> proportionality.
> 
> I think perfect geographical proportionality would violate perfect 
> political proportionality, so we can only provide approximate 
> geographical proportionality if political proportionality is a must.
> 
> Let's take a basic closed list method. First we will count the exact 
> proportionality split between the parties. Then we will (in theory) 
> check all possible combinations of candidates that respect the agreed 
> political proportionality split. Out of these we could elect e.g. the 
> one where the average distance to the nearest representative is lowest.

Here's one idea, based on the opinion reconstruction ideas I've been 
having the last few days. The ballot format is closed party list PR. 
First figure out how many seats each party is entitled to, which can be 
done using Sainte-Laguë. Call the number of representatives p. Then 
construct an estimate of the probability function over all locations in 
the country, that a random voter there votes for the party - this can be 
done using kernel density estimation if I'm not mistaken. Finally, 
choose a set of p from the number of candidates running on the party 
list so that the difference between the estimate made using population 
data and the estimate made using only the points for those candidates is 
minimized. Elect that subset, which, using this method, should be the 
candidate subset whose geographical distribution most closely 
approximates the geographical distribution of the voters.

There are two problems with this idea (three if you count the 
computational intractability of trying all subsets). The first is that 
the party always gets the number of seats it "deserves" on a national 
basis, meaning that if all the candidates are clustered around the 
capital, there's no incentive to spread out because all the subsets will 
be equally bad (and thus equally good). That could perhaps be handled by 
not listing candidates more than a certain distance (or number of 
voters) away from the voter, so that having too distant candidates makes 
the party incur an effective vote penalty. If one does so, however, even 
a perfectly estimated probability distribution will differ from the real 
probability distribution.

The second is the devil in the details of kernel density estimation: 
it's not obvious to me how one would pick the bandwidth. Would it have 
the same bandwidth for the estimation based on the candidate set as that 
based on the population set (because candidates are people as well), or 
different bandwidths? And if so, how does one determine the optimal 
bandwidth choice for two-dimensional KDE? I know too little about kernel 
density estimation to answer this.

For this method, the similarity measure would probably be one of the 
good proportionality measures as mentioned in 
http://www.votingmatters.org.uk/ISSUE20/I20P4.PDF : root-mean-square, 
Sainte-Laguë index, Gini disproportionality measure, Loosemore-Hamby 
index, or Monroe index.

It would be very hard to generalize this idea to party-neutral PR, but 
if the method works by reconstructing opinion space based on 
preferences, then the geographical distribution could be added as yet 
another dimension (or rather, two). Political/geographical balancing 
would then be done by artificially smoothing the data on the 
geographical dimensions, thus dampening both political 
disproportionality as well as the incentive towards parochialism.